DRDC-RDDC-2014-P78
Nonuniform Deployment of Autonomous Agents in Harbor-Like
Environments
Suruz Miaha , Bao Nguyena , Alex Bourquea , Davide Spinellob
a
Defence Research and Development Canada Center for Operational Research and Analysis, 101 Colonel By Drive
Ottawa, Ontario, K1A 0K2, Canada,
E-mail: {Suruz.Miah, Bao.Nguyen, Alex.Bourque}@drdc-rddc.gc.ca.
b
Department of Mechanical Engineering, University of Ottawa, Ottawa, Ontario, K1N 6N5, Canada,
E-mail: [email protected].
We propose a nonuniform deployment strategy of a group of homogeneous autonomous agents in harbor-like environments. High
value units berthed in the area need to be secured against external attacks. Defenders deployed in the area are expected to monitor,
intercept, engage, and neutralize threats. In the framework of decentralized coordinated multi-agent systems, we model and simulate
the optimal deployment of a group of mobile autonomous agents that accounts for a risk map of the area and the optimal trajectories
that minimize the energy consumed to intercept a threat in a given area of interest. Theoretical results are numerically illustrated
through simulations in a realistic harbor protection scenario.
Keywords: nonuniform deployment; threat interception; harbor protection.
1.
Introduction
dressed in [40–42], and random link failures [43,44]. Here, in
order to focus on the optimal deployment and interception
with non-uniform risk, we assume that full communication
exists among agents at all times, leaving the treatment of
underlying time-varying communication networks to current and future work. This work contributes to the development of optimal strategy to deploy mobile autonomous
agents and intercept threats in a harbor–like environment
assuming that agents coordinate in a decentralized fashion by sharing information between them, without the intervention of an external unit. The area of deployment is
assumed to be small enough so that information sharing
capabilities allow each agent to communicate with the others in the group. Specifically, the main contributions of this
paper are a set of possible solutions to address the following
questions:
Cooperative missions with mobile sensor networks are being studied extensively from algorithmic to implementation
aspects. Mobility, coupled with estimation, data fusion, and
motion control algorithms, enables groups of mobile agents
to coordinate in real-time to achieve specific objectives.
These ideas have been applied to a variety of domains,
including threat tracking [1–13], formation and coverage
control [6, 14–25], rendezvous and deployment [26], environmental tracking and monitoring [27–30], and resource
allocation problems with heterogeneous entities [31]. Comprehensive reviews can be found in [32–34], and several
technical aspects are discussed in the book [35]. Here, we
extend some of these ideas to securing harbors and ports
against underwater threats, which is a problem with many
opened challenges [36]. Securing ports against threats is especially acute when they are located near strategic choke
points, such as the Strait of Hormuz, the Strait of Gibraltar, the Suez Canal, and the Panama Canal [36]. Among
the most significant challenges in harbor protection study is
that the visibility of both the attackers and agents is likely
to be poor during their mission. In many realistic scenarios, sensory measurements are noisy and communication in
the underwater domain is intermittent, leading to modeling frameworks similar to the one presented in [37] to study
synchronization of intermittently coupled continuous-time
nonlinear oscillators, with mean square convergence rates
obtained in a discrete-time setting in [38, 39], and several
extensions that include stochastic consensus protocols ad-
i) How and where to deploy agents throughout the harbor so that the likelihood of defending against threats
is maximized?
ii) Given the agent’s safe stand-off distance, how to intercept a threat in pre-specified reaction time while minimizing agent’s actuator energy?
Deployment strategies developed here are of general interest in the broad field of cooperative missions with mobile
multi-agent systems. However, they are illustrated herein
for a harbor defence system taking into consideration a
realistic harbor geometry, so that the abstraction of the
formulation is immediately translated into suitable scenarios that facilitate the understanding of the applicability of
1
DRDC-RDDC-2014-P78
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S. Miah, B. Nguyen, A. Bourque, D. Spinello
the proposed methods and solutions. It is assumed that the
threats are initially outside the harbor’s outermost boundary and the agents are placed closed to sensitive areas to be
protected. The defence system is structured in two stages,
where the first stage consists of deploying multiple mobile
agents based on the risk distribution of the harbor. The
risk zones inside the harbor are assumed to be known a
priori, and they are encoded by a risk distribution with
eventual non-uniformities. The area is then spatially partitioned according to Voronoi partitioning technique such
that each Voronoi cell is assigned to a single agent. Like
that, agents move in a fully decentralized fashion in the
sense that each agent receives the position information from
all other agents while employing its actions independently.
Given the maximum sensor range, we determine the minimum number agents that needs to be deployed throughout
the harbor. In the second stage, each agent will intercept
the threats inside its own Voronoi cell taking into consideration its safe stand-off distance.
Yip et al. in [45] focused on formulating a stochastic model of the effectiveness of a harbor defence system
against underwater threats. Their model is based on the
assumption that only one agent is deployed against a single
threat inside the harbor. In addition, the interception technique described in [45] assumes that the threat’s trajectory
is simply a straight line, which may be a too restrictive
or non realistic hypothesis to model relevant real scenarios. Several research works have been conducted to protect
harbors (civilian and military) and ports against underwater threats, see [46–51], for example, and some references
therein. Authors in [52] presented a system of organic sensors coupled with one or more agents to protect high-value
assets, where fuzzy logic algorithms were developed to determine priorities of both underwater and surface threats to
be neutralized. However, a central control unit is required
for their scenario to operate in real-time. Nevertheless, the
nonuniform deployment of agents based on harbor’s risk
level are not considered in the literature to date. It turns
out that the optimal deployment of agents and intercepting
threats with minimum actuating energy are still among the
main challenges, which we address in this paper. It is important to point out the fact that the proposed deployment
strategy in a harbor defence system can be applied to scenarios where multiple defenders need to neutralize multiple
threats. The work described herein exploits the concepts of
multi-agent systems [53], the optimal discrete-time control
and estimation theory [54].
The rest of the paper is outlined as follows. Section 2 gives the problem description and assumptions made
throughout the paper including a high-level architecture of
a harbor protection scenario. Section 3 details how to optimally deploy a set of homogeneous agents inside the harbor’s inner reaction zone while maximizing coverage metric
taking into account the risk maps. Section 4 describes how
the agents optimally intercept a threat while minimizing
their actuating energy. Section 5 presents the main theoretical results through computer simulations that encode
the modelling assumptions. Section 6 gives the summary,
conclusions, and potential future research avenues.
2.
Assumptions, Harbor Architecture, and
Modeling
Throughout this paper, scalar quantities will be denoted by
lower-case letters, while vectors will be denoted by lowercase bold letters. Upper-case bold letters will denote matrices. For any positive integer n, Rn denotes the Euclidean
space.
2.1.
Assumptions
The current work relies on the following preliminary assumptions:
• Agents and threats operate in a 2D submerged
workspace in the harbor.
• Agents can perfectly (without noise) communicate
among them and each agent is responsible to intercept threats in its own zone of the harbor.
• Agents can estimate their position and velocity
without noise.
• The harbor infrastructure is stationary.
The terms agent, interceptor, and defender will be used
interchangeably throughout this paper.
2.2.
Harbor System Architecture
A high-level architecture of a harbor defence system is depicted in Fig. 1. The shaded with zigzag pattern is the high
value unit, that is located at the left of the harbor area. The
presence of the high value unit introduces non-uniformity in
the risk zone, and therefore it allows us to treat this important case. Sensors (sonars, for example) are mounted on the
outermost trip-wire boundary. The purpose of the trip-wire
sensors is to detect and track attackers while they are in
the detect and track zones (see Fig. 1). In the scenario illustrated in Fig. 1, four agents are deployed inside the harbor
area (inner reaction zone). As can be seen, the threats may
enter from several directions (underwater or sideways, for
instance) around the harbor. As such, the risk probability
(which depends on the frequency of attacks by the threats)
is also nonuniform. We consider a decentralized architecture in which there is no central control unit, as opposed
to the scenario presented in [52]. Agents’ optimal configurations in the inner reaction zone maximize a nonuniform
coverage metric. In the presence of threats inside the inner
reaction zone, each agent intercepts the threats that are
inside its own Voronoi cell in the inner reaction zone of the
harbor.
2.3.
Motion Models of Agents and Targets
We consider a group of n homogeneous autonomous agents.
Let γ and k denote the discrete sampling time interval and
DRDC-RDDC-2014-P78
Nonuniform Deployment of Autonomous Agents in Harbor-Like Environments
3
Trip−wire sensors (e.g., sonars)
Target
Outermost boundary
Bridge
1.5 km
Detect and track zone
100 m
s
r
d
20 m
Autonomous agents
2.0 km
HVU Reference point
Inner Reaction Zone (IRZ)
(danger zone)
Damage contour (innermost boundary)
2.5 km
Underwater Barrier (UWB)
Harbour entrance
Detect and track zone
Island
Trip−wire sensors (e.g., sonars)
Outermost boundary
Targets
Fig. 1.
Harbor protection system architecture.
discrete time index, respectively, characterizing the evolution of the agents in discrete time. The discrete time motion
model of the ith, i ∈ {1, 2, . . . , n} ≡ I, agent is given by
the following set of equations that describe the linearized
kinematics of an accelerated material particle
ṗ[i] = u[i]
[i]
(1a)
[i]
=α .
(1b)
T
where p[i] (t) = x[i] (t), y [i] (t) is the 2D position, v[i] =
T
[i]
[i]
vx (t), vy (t) is the velocity at time t ≥ 0, and u[i] and
v̇
α[i] are the vectors of velocity and acceleration inputs, respectively. The kinematic model (1) approximates the 2D
trajectory of the ith agent when it is required for agents
to converge to a desired position with velocity determined
by (1b).Similar to the agent model (1), we assume that
each threat moves according to the linearized accelerated
kinematics
ṡ = u
v̇ = α
(2a)
(2b)
where s and v are the threat’s 2D position and velocity
vectors, respectively, with the corresponding 2D actuating
inputs u and α.
3.
generalized coverage metric that include the nonuniform
risk map and the motion of the agents. The purpose of
this section is to present motion coordination algorithms
to optimally place agents. For that, we formally define the
network model for the agents and the Voronoi tessellation
of the inner reaction zone of the harbor.
Deployment of Agents
The agents are distributed in optimal locations inside the
harbor, where the optimality is defined with respect to a
3.1.
Harbor Partitioning and Network Model
Let p = (p[1] , . . . , p[n] ) be the collection of 2D positions of
agents’ network in the 2D inner reaction zone, Q ⊂ R2 , of
the harbor. For optimal spatial placements and area coverage, agents partition the area to be covered using the generalized Voronoi partitioning technique [31, 55]. The Voronoi
partition of the harbor is denoted by
V(p) = (V1 (p), . . . , Vn (p)),
where the ith Voronoi cell is defined by
Vi (p) = {q ∈ Q : f (ri ) ≥ f (rj ), ∀j ∈ I \ {i}} ,
∀i ∈ I, ri = q − p[i] is the Euclidean distance, and f (·) is
the sensor performance function that will be defined in subsection 3.3. Therefore, the generators of the Voronoi partition are the states (p[1] , . . . , p[n] ). For simplicity, Vi (p)
will be denoted by Vi throughout the paper. Intuitively,
Vi represents an area where each point is better sensed by
the ith agent than to all other agents. Interested readers
are referred to [55] for the comprehensive study on Voronoi
partitioning and its applications.
DRDC-RDDC-2014-P78
4
S. Miah, B. Nguyen, A. Bourque, D. Spinello
A network of agents in the inner reaction zone Q ⊂ R2 ,
is defined by the
tuple (I, R, E), where
R ≡ set of agents
i
[i] [i]
[i]
i
P , U , P0 , f
, with P [i] ⊂ R2
≡ R i∈I =
given by
i∈I
i
is the state space of the ith agent, U is the set of con[i]
trol actions (accelerations) that define the trajectories, P0
is the set of initial states, and f [i] is the physical state
transition function of the ith agent. The communication
n
edge map is given by E : i=1 P [i] → I × I. The physical state and the initial physical state of the ith agent
[i]
[i]
are p[i] ∈ P [i] and p0 ∈ P0 , respectively. We refer to
n
p = (p[1] , . . . , p[n] ) ∈ i=1 P [i] as a state of the agents’
network in the harbor’s inner reaction zone.
Note that the map p → (I, E(p)) models the topology
of the communication network among the agents. Hence,
(I, E(p)) denotes the communication graph. Two agents at
locations p[i] and p[j] communicate if and only if the pair
(i, j) ∈ E(p). The informal description of our deployment
strategy for agents inside the harbor can be schematized as
follows:
φ(q)dΩ,
Mφ (Vi ) =
Vi
1
Cφ (Vi ) =
qφ(q)dΩ,
Mφ (Vi ) Vi
(4)
(5)
respectively, with dΩ being the area element.
3.3.
Nonuniform Deployment
Motivated by the locational optimization problem [55], we
consider a generalized total coverage metric in terms of total power (signal strength) received by agents
n φ(q)f (ri )dΩ
(6)
H(p, V) =
i=1
Vi
where ri = q − p[i] , ∀q ∈ Vi . Note that f (·) is the agent’s
sensor performance defined by
f (ri ) = k exp(−βri2 )
Step 0: Set up a communication protocol, such as agents
will communicate with each other at discrete time instants during their entire mission. The ith agent performs Step 1 and Step 2 for each communication round:
Step 1: It broadcasts its own message (2D position and
unique ID) to all other agents and receives the set of
messages from all other agents.
Step 2: It updates its logical state (determining geometric centers, for example), which will be detailed later, of
its own Voronoi region Vi inside the harbor.
Step 3: It updates its physical state p[i] by moving towards the geometric center of Vi before the next communication round.
3.2.
Modeling Risk Zone
Note that there might be different risk zones through which
threats are likely to attack. Inspired by [56], we define risk
zones in the harbor as the area where threats frequently enter to damage the high value unit. Following [17], we define
the risk function φ : P −→ R+
0 as
− 12
φ(q) = e
T
(qx −q̄x )2
2
σx
+
(qy −q̄y )2
2
σy
,
(3)
where q = [qx , qy ] ∈ Q is the position vector. Model (3)
defines the risk zone centered around the location (q̄x , q̄y )
with standard deviations σx and σy along the X-axis and
Y -axis, respectively. Note that the highest risk is at the location (q̄x , q̄y ). Since the risk function is nonuniform, this
method leads to nonuniform distribution of agents, that are
more densely deployed in areas with higher risk. The mass
Mφ (Vi ) and the centroid Cφ (Vi ) of the Voronoi cell Vi are
with k > 0 and β > 0. The metric (6) encodes how rich
the coverage provided by the agents’ network in Q is. In
other words, higher H implies that the corresponding set
of agents achieves better coverage of the area Q. Commonly
the Voronoi tessellation problem with mobile generators is
formulated by modeling the dynamics of the agents as simple integrators, and therefore the coverage metric can be
extremized by the well–known Lloyd algorithm [57]. Here,
by considering accelerated particles taking into account the
acceleration inputs, we want to maximize the coverage metric H as defined in (6), while the speeds of agents governed by (1b) reach zero at the optimal configurations of
agents that eventually maximize the coverage. Note, however, that agents may reach optimal configurations before
the acceleration inputs and velocity of the ith agent decay
to zero as t → ∞. An appropriate feedback law, defined in
equation (11), for each of the input vectors in (1) dictates
that each agent moves towards its generalized centroid with
speed greater than the speed generated by the acceleration
inputs. This guarantees that the system dynamics does not
violate basic physics principles encoded by Newton’s second
law.
Using the results presented in [31, 53], the derivative
of Hi in (6) with respect of p[i] yields
∂H [i]
[i]
=
M
(V
)
C
(V
)
−
p
,
i
i
φ̃
φ̃
∂p[i]
where φ̃(q, p[i] ) = −2φ(q) ∂f (ri )/∂(ri2 ) ,
φ̃(q, p[i] )dΩ,
Mφ̃ (Vi ) =
V
i
1
Cφ̃ (Vi ) =
qφ̃(q, p[i] )dΩ.
Mφ̃ (Vi ) Vi
(7)
(8)
(9)
DRDC-RDDC-2014-P78
Nonuniform Deployment of Autonomous Agents in Harbor-Like Environments
It is worth pointing out that φ̃(q, p[i] ) can be interpreted
as modified risk levels that includes the agents’ sensor performance in the harbor region Vi , which is occupied by
the ith agent and its corresponding mass and centroid are
Mφ̃ (Vi ) and Cφ̃ (Vi ), respectively. From the above definition it follows that φ̃ > 0 since f is strictly decreasing in
its argument.
The standard Lloyd descent algorithm [58] states that
a necessary and sufficient condition for the agents to be
in the optimal configuration that maximizes the coverage metric (6) (with constant density φ) is to asymptotically converge to the centroids of the Voronoi tessellation, i.e., P = {p[i] (t)|p[i] (t) = Cφ (Vi )(t)} as t → ∞.
Hence, if the agents (with kinematics (1)) are asymptotically deployed to the corresponding centroids, Cφ̃ (Vi ), of
the Voronoi tessellation, then the coverage metric is maximized as it is clear from (7) and therefore the configuration
is optimal. Feedback laws that map to optimal trajectories of the closed loop system associated to the dynamics
(1) are given in the following proposition, by showing that
these trajectories belong to the largest invariant set of a
quadratic Lyapunov function.
Let κ and μ2 be positive constants. Before defining
the feedback laws for the agent dynamics (1), we introduce the following assumption which relates tracking errors with respect to Voronoi centroids to actuation speed,
posing that the actuation speed v[i] cannot vanish before
the ith agent converges to the centroid of the corresponding
Voronoi cell.
Assumption 3.1.
v[i] ≥ κ̄Cφ̃ − p[i] ,
(10)
for some κ̄ > 0.
With the following proposition we give a feedback
law for asymptotic convergence towards the centroids, and
therefore optimal coverage in the sense of (6).
Proposition 3.2. Consider a group of n agents with dynamics (1). Let the function H defined in (6).
(1) Suppose that Assumption 10 holds ∀t ≥ 0, the trajectories of the closed–loop system generated by
the feedback laws (assuming Cφ̃ = p[i] )
u[i] = v[i] Cφ̃ − p[i]
Cφ̃ − p[i] .
α[i] = −κv[i] , κ > 0.
(11a)
(11b)
asymptotically drive the agents to the generalized
centroids (9) with zero velocity, that is
lim p[i] (t) = Cφ̃ ,
t→∞
lim v[i] = 0.
t→∞
(2) The trajectories of the closed loop system associated to the feedback law (11) are optimal with
respect to the metric (6).
Proof.Let
J=
n 1
i=1
2
1
μ1 v + μ2 Cφ̃ − p[i] 2
2
[i] 2
5
(12)
be a candidate Lyapunov function (cost function). With
the substitution of feedback laws (11) the time derivative
of the J along the trajectories is given by
n 2
−μ1 κv[i] − μ2 v[i] Cφ̃ − p[i] (13)
J˙ =
i=1
which is negative definite for nonzero positive constants μ1
and μ2 . By using 10 the following bound follows
n 2
−μ1 κv[i] − μ2 v[i] Cφ̃ − p[i] J˙ =
i=1
≤
n i=1
2
2
−μ1 κv[i] − μ2 κ̄Cφ̃ − p[i] <0
(14)
for v[i] = 0 and p[i] = Cφ̃ . By the global LaSalle’s principle it follows that the asymptotically all trajectories are
attracted into the largest invariant set of J˙ = 0, which implies point (1) since the largest invariant set includes only
v[i] = 0 and p[i] = Cφ̃ .
The proof of point (2) is completed by using a classical
Lloyd algorithm argument, according to which p[i] −Cφ̃ = 0
is the largest invariant set of the coverage metric H that is
maximized along the trajectories associated with the feedback law u[i] , since such trajectories are proportional to its
gradient ascent directions (see equation (7)).
Note that the chain of inequalities (14) without the
additional condition (10) would imply J˙ = 0 simply for
v[i] = 0; therefore at the equilibrium the agents would
not be required to converge to centroids of the Voronoi
cells. Therefore, by requiring through condition (10) that
the actuation velocity does not vanish before convergence
to Voronoi centroids we also guarantee that equilibrium
conditions correspond to optimal coverage.
4.
Intercepting Threats
If threats leak through the underwater barrier, it is the
agents’ responsibility to intercept and engage them before
they reach the damage contour of the high value unit. For
that, an agent has to perform the following two major steps
before engaging:
• Estimating threat’s position and velocities.
• Intercept threats.
We consider on board sensory apparatuses taking noisy
measurements of threats’ trajectories at discrete times. Between two measurement times the threats’ dynamics are approximated by a linearized stochastic process arising from
the finite difference of (2)
g[k + 1] = Φg[k] + Γ (α[k] + ω[k]) ,
(15)
DRDC-RDDC-2014-P78
6
S. Miah, B. Nguyen, A. Bourque, D. Spinello
where
⎡
⎤
⎡
(2) A posteriori estimate (correction): the Kalman
gain is computed by
⎤
2
10γ 0
γ /2 0
⎢0 1 0 γ ⎥
⎢ 0 γ 2 /2⎥
Φ=⎣
, Γ=⎣
,
0 0 1 0⎦
γ
0 ⎦
0001
0
γ
K[k + 1] = P− [k + 1]H̄T [k + 1]
(H̄T [k + 1]P− [k + 1]H̄T [k + 1]+
Σ[k + 1])−1 ,
and γ is the sampling time. The state g[k] =
T
[x[k], y[k], vx [k], vy [k]] includes 2D position and velocity at time instant k; α[k] = [αx [k], αy [k]] is the 2D acceleration input at time instant k, R2 ω[k] ∼ N (0, Λ)
is the process noise (white, zero-mean Gaussian random
sequence) with Λ being the noise covariance matrix. The
expected values of the threat’s initial state and its error
covariance are given by
where H̄[k + 1] is the Jacobian with respect to the
state g of the non-linear measurement model h(·)
in (16). The threat’s state estimate is updated with
current measurements using
ĝ+ [k + 1] = ĝ− [k + 1]+
K[k + 1] z̄[i] [k + 1] − h(ĝ− [k + 1], 0)
E(g[0]) = ĝ[0],
T
E (g[0] − ĝ0 ) (g[0] − ĝ0 ) = P[0].
The error covariance associated with the a posteriori estimate is updated by
P+ [k + 1] = (I4 − K[k + 1]H[k + 1])P− [k + 1].
Since the ith interceptor measures the line-of-sight (LOS)
distance between itself and the threat at time instant k,
the measurement model can be written as
z̄[i] [k + 1] = h (g[k + 1], ξ) = d[i] [k + 1] + ξ[k + 1], (16)
[i]
[i]
[i]
where d[i] [k] = [d1 [k], . . . , dN [k]]T with dι [k] being the
Euclidean distance (LOS) given by
d[i]
ι [k + 1] =
(x[i] [k + 1] − xι [k + 1])2 + (y [i] [k + 1] − yι [k + 1])2 ,
N is the number of LOS distance measurements,
ι = 1, . . . , N , (xι [k], yι [k]) is the 2D position of the threat
[i]
corresponding to the measured distance dι , and RN ξ[k] ∼ N (0, Σ) is the measurement noise vector with zeromean and Σ measurement noise covariance matrix. Having multiple line–of–sight (LOS) distance measurements between the ith interceptor and a threat at time instant k, the
state ĝ[k] of a threat can be estimated using the Extended
Kalman Filter (EKF) algorithm [59], since measurements
are nonlinear functions of the state. The EKF employs the
following two main phases at each time step for the interceptor to estimate the threat’s state:
(1) A priori estimate (prediction): the threat’s new
state at time step k + 1 is predicted using:
ĝ− [k + 1] = Φĝ+ [k] + Γα[k], ĝ+ [0] = ĝ[0], (17)
where the subscripts “− ” and “+ ” denote the a
priori and a posteriori estimates, respectively. The
error covariance matrix associated with this prediction is computed by:
T
T
P− [k + 1] = ΦP+ [k]Φ + ΓΛΓ ,
P+ [0] = P[0].
where I4 is the four dimensional identity matrix.
We assume that interceptors can estimate their position and velocity without noise. Similar to the model (15),
the dynamic model (1) for the ith agent can be approximated in discrete–time by the linearized finite difference
kinematics
(18)
[i] [k + 1] = Φ[i] [k] + Γα[i] [k],
T
[i]
[i]
where the state [i] = x[i] [k], y [i] [k], vx [k], vy [k] represents the state vector of 2D position and velocity at time
[i]
[i]
instant k, and α[i] [k] = [αx [k], αy [k]] being the 2D acceleration input at time instant k. Agents use the threat’s
estimated state ĝ+ [k] to intercept, and the evolution of the
error is a tracking measure. By subtracting (17) from (18)
we obtain the following propagated error in discrete time
[i]
[i]
e− [k + 1] = Φe+ [k] + γ α̃[i] [k],
(19)
[i]
where e+ [k] = [i] [k] − ĝ+ [k] that is computed through
EKF’s prediction and update steps at each time instant k
and α̃[i] [k] = α[i] [k] − α[k]. Given the reaction time tr , the
problem is to determine the optimal acceleration history
[i]
[i]
(ux [k], ux [k]) (inputs to the agent) such that the agent
can intercept the threat while maintaining its safe stand-off
distance, rw , before the threat reaches the damage contour
around the high value unit. The detail derivations for gen[i]
[i]
erating (ux [k], ux [k]) are adapted from [60, Ch. 3] and are
given in the Appendix 1.
5.
Numerical Results
In this section, we provide a set of computer simulations
to demonstrate the capability of a team of mobile agents
in maximizing the probability of detecting a certain event
in a harbor environment when individuals trajectories are
DRDC-RDDC-2014-P78
Nonuniform Deployment of Autonomous Agents in Harbor-Like Environments
generated by the feedback law (11); in this case the performance metric is the nonuniform coverage metric defined
in (6) which can be interpreted as the probability of detecting an event in a given area. In addition, we show how an
agent intercepts a mobile threat within its own region in
pre-specified reaction time while minimizing its actuating
energy; in this case the performance measure is the position
error (generated from the discrete time error model (19))
between an agent and a mobile target, while maintaining a
safe stand-off distance. All simulations (both agent deployment and intercepting threats) are performed in discrete
time with a sampling time interval γ = 0.05 s.
5.1.
Deployment
The optimal coverage of the agents’ nonuniform deployment in the harbor is illustrated by simulating a 2D inner reaction zone of the harbor with its boundary vertices
at (0.02, 0.02), (1.2, 0.2), (2, 0), (2.2, 1), (2, 2.5), (0.6, 2.7),
(0, 2), (0.05, 1.6), and (0, 1) km. The distribution of risk
zones throughout the harbor is given by (3), where the
highest risk is in the region centered around (x̄, ȳ) with
σx = 0.6 km and σy = 0.4 km. It implies that threats can
enter most-likely through regions centered around (x̄, ȳ) =
(1.5, 0.2), (0.4, 0.15), (1.2, 2.5) km. Initially, 15 (n = 15)
agents (hollow arrows with unique IDs i = 1, . . . , 15) are
placed close to the high value unit located at the middle of
the left side of the harbor. The heading direction of each
agent is toward the centroid of its corresponding Voronoi
partition as computed by (5). The colormap on the right
of Fig. 2 shows corresponding value of the risk level in the
harbor, where 0 (zero) and 1 (one) correspond to lowest
and highest risk level, respectively.
The evolution of agents’ network at time t ∈ [0, 60] s
is determined through models (1) and (11) for i = 1, . . . , n,
and κ = 1. Their corresponding total coverage is computed
by the coverage metric given in (6). The optimal configuration of the agents in total readiness time 60 s is revealed
in Fig. 2 with the maximum coverage by the agents. Fig. 3
shows the normalized total coverage. The agents yields the
maximum coverage of the harbor, which corresponds to
unity (see Fig. 3). As expected, agents speeds decreasing
while they move towards their optimal configurations. This
model naturally leads to nonuniform deployment as dictated by the risk function, with higher risk zones more
densely populated than lower risk zones. Nine agents are
densely placed close to the harbor entrance and six agents
are deployed from across the harbor entrance since there is
a high probability of entering threats from these locations
of the harbor.
5.2.
Intercepting Threats
Assume that the agents are in optimal configurations as
shown in Fig. 2(b). Here we show how the agent 14, for
instance, intercepts a threat using its optimal acceleration
inputs α[14] that minimize the actuating energy modeled
7
by the cost function (A.1) while maintaining its safe standoff distance. Since the harbor area that we have chosen is
in the range of kilometer, for the purpose of showing the
trajectory in meters we magnify the Voronoi cell of agent
14. For the proof of concept, we assume that the agent 14 is
initially placed in the optimal loiter position (60 m, 10 m)
in the inner reaction zone of the harbor. An alert has been
sent to all agents that a threat is heading from the position (98, −8) m towards the high value unit with an initial
velocity of 1.2 m·s−1 (≈ 2.3 kn). Since the threat will surely
be entering through Voronoi cell of agent 14, the agent 14
moves according to the kinematic model (18), where its
acceleration inputs are computed by (A.8). Figs. 4(a) and
4(b) show their initial positions and velocities, respectively.
The agent is always pointed towards the threat. The initial
position error is about 42 m as shown in Fig. 5. Since the
agent has to intercept the threat before it reaches the high
value unit, it drastically increased the speed to ≈ 2.7 m·s−1
(≈ 5.2kn) in about 4 s as shown in Fig. 4(b). Once the
agent’s position is about 15 m (safe stand-off distance, rw )
away from the threat, it maintains the same speed as the
threat. This is clear from Fig. 5.
One important observation that can be made from this
simulation results is that if the threat’s speed is about 4 kn
and it is initially about 42 m away from the agent, then it
can intercept the threat in about 10 s and start engaging
with its available weapons, for instance.
6.
Summary and Conclusion
In this paper, we focus on an optimal deployment of multiple autonomous agents with the purpose of protecting a
high value unit against multiple attacks in a harbor-like
environment. A nonuniform risk distribution function is assigned to the area to be protected and agents are optimally
distributed throughout the harbor to maximize a generalized coverage metric that encodes the non uniform risk
distribution and the performance of sensory devices in detecting events in the area where they are deployed. The
coverage metric is maximized along the trajectories generated by a feedback law with kinematic actuating actions.
Agents are deployed more (less) densely in higher (lower)
risk zones using Voronoi tessellation. Once threats penetrate the underwater barrier, the mobile agents are notified
to intercept them. If a threat reaches in one of the Voronoi
cells, the corresponding agent estimates the threat’s position and velocities and intercept it in pre-specified reaction
time. This is performed using the extended Kalman filter
and the optimal linear quadratic control technique, which
minimizes the agent’s actuator energy that need to be applied to intercept the threat in its Voronoi cell.
Even though the current work focused in deploying
agents patrolling in a harbor protection environment, the
theoretical results are applicable to a general class of coverage optimization problems, such as nonuniform distribution of food supply among a group of animals, collecting
scientific data in an underwater environment, and optimal
DRDC-RDDC-2014-P78
8
S. Miah, B. Nguyen, A. Bourque, D. Spinello
risk
risk
1
3
0.9
2.5
0.9
2.5
0.8
4
2
2
0.7
3
2
2
7
5
Y [km]
7
8
0.5
9
10
11
12
13
14
15
1
0.5
0.6
Y [km]
0.6
6
HVU
0.7
3
4
1.5
0.8
5
1
1.5
1
HVU
6
0.5
9
1
0.4
0.3
10
0.3
0.5
12
13
14
0
0.4
8
15
0.2
11
0.2
0
0.1
−0.5
−0.5
1
3
0
0.5
1
1.5
2
0.1
−0.5
−0.5
0
2.5
0
0.5
X [km]
1
(a)
Fig. 2.
1.5
2
2.5
0
X [km]
(b)
Nonuniform coverage: agents’ optimal configurations; colorbar represents risk level inside harbor
.
1
0.9
normalized coverage
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
10
20
30
40
50
60
Time [sec]
Fig. 3.
Performance of the agents’ nonuniform deployment.
configurations of a team of agents for estimating a target
location.
The current harbor protection scenario assumes that
agents are able to communicate among them perfectly
(noise-free) without any information loss. However, this is
not the case when a harbor contains islands or obstacles
yielding intermittent communications among agents while
accomplishing their missions. The optimal deployment of
agents in the presence of such intermittent communications
still remains a significant challenge and is a potential future
research avenue, which is currently under investigation.
Acknowledgments
The authors would like to thank anonymous reviewers for
their constructive feedback on this manuscript.
Appendix A Generating Optimal Acceleration Inputs for ith Agent
DRDC-RDDC-2014-P78
Nonuniform Deployment of Autonomous Agents in Harbor-Like Environments
100
9
3
target
interceptor
90
2.5
80
70
Velocity [m/sec]
Y [m]
60
50
40
30
2
1.5
1
20
10
0.5
target
interceptor
0
−10
−10
0
10
20
30
40
50
60
70
80
90
0
100
X [m]
0
10
20
30
(a)
Fig. 4.
40
50
60
Time [sec]
(b)
Intercepting a threat: (a) trajectories and (b) speed history; of the agent 14 and the threat.
45
Position error [m]
40
35
30
25
20
15
10
agent’s safe stand−off distance
5
0
0
10
20
30
40
50
60
Time [sec]
Fig. 5.
Position error between the agent and the threat.
In this section, we will derive the optimal acceleration inputs αi [k], k = 0, . . . , kf − 1, for the ith,i = 1, . . . , n,
agent to intercept a threat within the reaction time tr . We
denote tk = γk as the time at discrete index k and kf
corresponds to the reaction time tr through tr = γkf . For
clarity, we simply drop the superscript [i] and the subscript
[i]
+ from the error vector e+ [k] and α̃ ≡ α̃[i] in the rest of
this section. We define ith agent’s cost function as
1
J(α̃) = eT (tr )P(tr )eT (tr )+
2
kf −1 1 tk+1 T
e (t)Q(t)e(t) + α̃T (t)R(t)α̃(t) dt, (A.1)
2
tk
k=0
where P, Q ∈ R4 are positive (semi) definite symmetric
matrices that represents the relative importance of position
and velocity error components between the agent and the
target. The matrix R is positive definite symmetric, which
imposes restriction on the amount of energy required for acceleration inputs α[k] (in terms of propeller thrust forces,
for example). The objective at this stage is to minimize
the cost J(α̃) for the ith agent. Since the agent can estimate the target’s position and acceleration (as illustrated
in section 4), the problem can be posed in terms of finding
α̃[k] ≡ α̃[i] that solves the following optimization problem:
inf J(α̃),
α̃k ∈R2
(A.2)
DRDC-RDDC-2014-P78
10
S. Miah, B. Nguyen, A. Bourque, D. Spinello
for k = 0, . . . , kf − 1.
The solution for α̃[k] of the problem (A.2) follows the
similar procedure illustrated in [60, Chapter 3]. We rewrite
the error (19) as
e(t) = Φ(t, tk )e[k] + Γ(t)α̃[k],
(A.3)
for t ∈ (tk , tk+1 ] and Φ(t, tk ) is the possible time–varying
state transition matrix. The cost functional (A.1) can be
written as
1 T
e (tr )P(tr )e(tr )+
2
kf −1 Q(t) 0
1 tk+1 T
e(t)
dt.
e (t) α̃T (t)
0 R(t) α̃(t)
2
k=0 tk
(A.4)
J(α̃) =
By considering a discrete time implementation, for t ∈
[tk , tk+1 ], α̃[k] we approximate α̃ as constant, that is
(i.e., α̃(t) = α̃[k], for t ∈ [tk , tk+1 ]). With this assumption and substituting e(t) from (A.3) in (A.4) yield
kf −1 1 tk+1
1 T
(
J(α̃) = e (tr )P(tr )e(tr ) +
2
2
k=0 tk
ΦT QΦ
T
ΦT QΓ
e[k]
T
dt,
e [k] α̃ [k]
(ΦT QΓ)T R + ΓT QΓ α̃[k]
(A.5)
with argument (t) dropped whenever no ambiguity arises.
Defining
tk+1
ΦT (t, tk )Q(t)Φ(t, tk )dt,
Q̂k =
tk
tk+1
M̂k =
tk
tk+1
R̂k =
ΦT (t, tk )Q(t)Γ(t)dt,
T
where Ck is the (2×4) optimal feedback gain matrix and Pk
is the solution of the following discrete-time matrix Riccati
equation
Pk = Q̂k + ΦTk Pk+1 Φk − M̂k + ΦTk Pk+1 Γk
(A.9)
−1 M̂Tk + ΓTk Pk+1 Φk
R̂k + ΓTk Pk+1 Γk
with Pk+1 = P[kf ] Pf .
For the agent to maintain a safe stand-off distance, the error vector ek , in (19) is replaced by ek +
[rw cos(π/4), rw cos(π/4), 0, 0]T , where rw is the agent’s safe
stand-off distance. The derivation of (A.8) and (A.9) follow from [60, Chapter 3]. The following steps summarize
the determination of the ith agent trajectory using optimal
acceleration inputs α[i] [k]:
Step 1: Define Q(t), R(t), and P(tr ) for the cost function (A.1).
Step 2: Compute Q̂k , M̂k , and R̂k using (A.6).
Step 3: Compute the matrix Pk backward using the
discrete-time matrix Riccati equation (A.9).
Step 4: Compute α̃k ≡ α̃[k] using the control law (A.8).
Step 5: Compute the agent’s optimal acceleration inputs
by α[i] [k] = α̃[k] + α[k], where α[k] is the targets acceleration input (assumed to be known).
Step 6: Generate agent’s state (position and velocity) trajectory by (18).
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